Skyscraper Sudoku Technique: How to Spot It, With Real Examples

A skyscraper is a single-digit pattern made of two strong links. Find two columns where a digit has exactly two spots each. If one end of each column sits in the same row (the base) and the other two ends (the tops) do not, then one of the tops must hold the digit, and any cell that sees both tops can lose it.

To spot one, pick a digit, look for columns (or rows) where it has only two candidates, and check whether two of those pairs line up on one end but not the other. It comes right after the X-Wing in most technique lists, and it turns up often once a hard puzzle stalls.

How the skyscraper works

Take the example above, from a real puzzle in our library, focusing on the digit 3.

• Column 4 has only two cells that can be 3: R1C4 and R8C4. One of them is the 3 of column 4.
• Column 9 also has only two: R2C9 and R8C9.
• R8C4 and R8C9 are both in row 8, so they cannot both be 3.

Now follow the two cases. If R8C4 is not 3, then R1C4 is. If R8C9 is not 3, then R2C9 is. At least one of the two base cells is not 3, so at least one of the tops, R1C4 or R2C9, is a 3. You do not know which, and you do not need to.

Any cell that sees both tops cannot be 3. R1C7 sees R1C4 along row 1 and R2C9 inside box 3. R2C5 and R2C6 see R2C9 along row 2 and R1C4 inside box 2. All three lose their 3.

Puzzle 201280 from the Sudoku Royale library. Givens (row by row, dots are blanks): ..1.........6..94....42..1...7.8.23..4.9..6..9...7.8..7365.2...528................ The candidates shown are what is left after singles, pointing and claiming, naked and hidden pairs, triples and quads, X-Wing and Swordfish, with nothing else applied.

How to spot a skyscraper

1. Work one digit at a time. Skyscrapers only ever involve one digit, so hide the rest in your head (or use a digit highlight if your app has one).
2. List the strong links. For that digit, note every column where it has exactly two candidates. Then do the same for rows.
3. Pair up the links. Two column links make a skyscraper when one end of each sits in the same row. Two row links make one when one end of each sits in the same column.
4. Check the tops. If the other two ends are also in the same row, you have an X-Wing instead. If they are not, look for cells that see both tops.
5. Eliminate. Where the targets are depends on the tops. If they sit in the same band of boxes, the targets are in the box of one top and the row (or column) of the other. If not, the only targets are the two cells where one top's row crosses the other top's column.

A second example: the sideways skyscraper

The pattern works just as well on its side, with two row links and a base in a column. This one is from a different puzzle, focusing on the digit 1.

Row 2 allows a 1 in R2C2 and R2C7 only. Row 7 allows it in R7C2 and R7C9 only. The left ends, R2C2 and R7C2, are both in column 2, so at most one of them is the 1. That forces a 1 onto R2C7 or R7C9.

Three cells see both tops. R1C9 shares box 3 with R2C7 and column 9 with R7C9. R8C7 and R9C7 share column 7 with R2C7 and box 9 with R7C9. Each loses its 1.

Column 2 has four candidates for 1 here, so the base is not a strong link. It does not need to be. The base cells only have to see each other, which is why a skyscraper can hide in a column that looks busy.

Puzzle 201555 from the Sudoku Royale library. Givens (row by row, dots are blanks): ....4......597..2.8.....9....9.24.1..34...8..2..15.3..3..2.576.9.................. The candidates shown are what is left after singles, pointing and claiming, naked and hidden pairs, triples and quads, X-Wing and Swordfish, with nothing else applied.

Skyscraper vs X-Wing

Both patterns start with two strong links on the same digit in parallel lines. The difference is the far end:

• In an X-Wing, both ends line up. The four cells form a rectangle and the digit is removed from the two cover lines.
• In a skyscraper, only one end lines up. The other end is offset, so the eliminations shrink to the few cells that see both tops.

That is why people call a skyscraper a broken X-Wing. If you already scan for X-Wings, you are one step away: every time two strong links almost form a rectangle but miss by one row, check for a skyscraper. It is also the simplest case of a finned X-Wing (the so-called sashimi form) and the shortest X-Chain: strong link, weak link, strong link.

Related single-digit patterns

The skyscraper belongs to a small family that all use two strong links on one digit plus one connection between them:

• 2-String Kite: one row link and one column link, joined in a box.
• Empty Rectangle: one strong link plus a box where the digit is confined to a cross.
• X-Chain: the general version, with as many links as you like.

For where these sit relative to pairs, fish and wings, see our list of sudoku techniques in learning order or the overview of advanced sudoku strategies.